assignment1sol - ma2176 a1 sol Functions, Limits and...

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1 ma2176 a1 sol Functions, Limits and Continuity 1. Find the largest possible domain of each of the following functions: (a) 5 4 2 ) ( 2 = = x x x x f y , (b) 2 25 ) ( x x f y = = , (c) 1 1 ) ( 2 = x x x φ Solution : (a) () 5 or 1 0 1 5 5 4 2 = = = + = x x x x x x The largest domain of ) ( x f { } \1 , 5 R =− (b) 5 5 0 5 5 0 25 2 + x x x x The largest domain of ) ( x f ] 5 , 5 [ = (c) 1 0 1 = = x x The largest domain of ) ( x { } R = 2. Let 3 2 2 , 1 fx x gx x =+ = . Find formulas for (a) ( ) f + , (b) g x f    , (c) ( ) g fx D , (d) ( ) f D and state their largest possible domains. Solution : (a) () () 1 2 2 ) )( ( 3 + + = + = + x x x g x f x g f The largest domain of ) )( ( x g f + is equal to the intersection of the largest domain of ) ( x f and the largest domain of ) ( x g . Since the largest domain of ) ( x f R = and the largest domain of ) ( x g { } R = , the largest domain of ) )( ( x g f + { } { } RR R =∩ = . (b) ) 2 ( 1 2 2 1 2 ) ( 3 3 + = + = = x x x x x f x g x f g The largest domain of ) ( x f g is equal to the intersection of the largest domain of ) ( x g and the largest domain of ) ( x f minus {} 0 : = x f x . Since the largest domain of ) ( x f R = and ( ) { } { } ( ) { } 3 / 1 3 2 0 2 : 0 : = = + = = x x x f x and also the largest domain of ) ( x g { } R = , the largest domain of ) ( x f g is { } ( ) { } 1/3 \ 2 , 2 R −∩ = . (c) 1 2 1 2 2 2 ) )( ( 3 3 3 + = + = + = = x x x g x f g x f g D
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2 Solve the equation 1 0 1 1 2 3 3 = = + = + x x x . The largest domain of () : is 1 2 ) )( ( 3 + = = x x f g x f g D \{ 1} R (d) 2 1 2 1 2 ) )( ( 3 + = = = x x f x g f x g f D The largest domain of 2 1 2 ) )( ( 3 + = x x g f D is: \{1} R 3. State whether each of the following is an odd function, an even function or neither. Prove your
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This note was uploaded on 11/09/2009 for the course ENG 91301 taught by Professor Lui during the Spring '08 term at Hong Kong Institute of Vocational Education.

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assignment1sol - ma2176 a1 sol Functions, Limits and...

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